1 Introduction

In 2018, the Large Hadron Collider beauty (LHCb) experiment measured the lifetime of \(\Omega _{c}^{0}\) using \(\Omega _{b}^{-}\rightarrow \Omega _{c}^{0}\mu ^{-}\bar{\nu }_{\mu }X\) decays and obtained [1]

$$\begin{aligned} \tau (\Omega _{c}^{0})=268\pm 24\pm 10\pm 2\ \textrm{fs}, \end{aligned}$$
(1)

which is roughly four times as large as

$$\begin{aligned} \tau (\Omega _{c}^{0})=69\pm 12\ \textrm{fs} \end{aligned}$$
(2)

in the Particle Data Group 2018 archive (PDG2018) [2]. In 2021, LHCb confirmed the previous measurement using \(\Omega _{c}^{0}\) produced directly from proton–proton collision [3], and in 2022, Belle II reported a similar result using \(\Omega _{c}^{0}\rightarrow \Omega ^{-}\pi ^{+}\) decays [4]. These recent measurements indicate that \(\Omega _{c}^{0}\) is not the shortest-lived weakly decaying charmed baryon, which conflicts with our previous understanding [5]. This demands theoretical explanation.

At present, the standard framework for understanding weakly decaying heavy flavor hadrons is heavy quark expansion (HQE) [6,7,8,9,10,11,12,13]. Under this framework, some attempts have been made to resolve the \(\Omega _{c}\) lifetime puzzle [14,15,16]. However, there is still a lack of more reliable calculation based on quantum chromodynamics (QCD) for the hadronic matrix elements of high-dimensional operators in HQE.

HQE describes inclusive weak decays of heavy flavor hadrons. It is a generalization of the operator product expansion (OPE) in \(1/m_{Q}\), and enables the systematic study of non-perturbative effects. The starting point of HQE is the following transition operator

$$\begin{aligned} \mathcal{T}=i\int \textrm{d}^{4}x\ T[\mathcal{L}_{W}(x)\mathcal{L}_{W}^{\dagger }(0)], \end{aligned}$$
(3)

where \(\mathcal{L}_{W}\) is the effective weak Lagrangian governing the decay \(Q\rightarrow X_{f}\). The total decay width of a hadron \(H_{Q}\) containing a heavy quark Q can be given as

$$\begin{aligned} \Gamma (H_{Q})=\frac{2\ {\textrm{Im}}\langle H_{Q}|\mathcal{T}|H_{Q}\rangle }{2M_{H}}, \end{aligned}$$
(4)

where \(M_{H}\) is the mass of \(H_{Q}\). The right-hand side of Eq. (4) is then calculated using OPE for the transition operator \(\mathcal{T}\) [5, 14]

$$\begin{aligned} 2\ {\textrm{Im}}\mathcal{T}= & {} \frac{G_{F}^{2}m_{Q}^{5}}{192\pi ^{3}}\xi \left( c_{3,Q}\bar{Q}Q+\frac{c_{5,Q}}{m_{Q}^{2}}\bar{Q}g_{s}\sigma \cdot GQ\right. \nonumber \\{} & {} \left. +\frac{c_{6,Q}}{m_{Q}^{3}}T_{6}+\cdots \right) , \end{aligned}$$
(5)

where \(\xi \) is the relevant Cabibbo–Kobayashi–Maskawa (CKM) matrix element, and \(T_{6}\) consists of the four-quark operators \((\bar{Q}\Gamma q)(\bar{q}\Gamma Q)\), with \(\Gamma \) representing a combination of Dirac and color matrices.

In fact, there was also a conflict between theory and experiment for the lifetime of \(\Lambda _{b}\) as early as 1996. Taking \(\tau (B^{0})=(1.519\pm 0.004)\) ps in PDG2022 [17] as a benchmark, for \(\tau (\Lambda _{b})=(1.14\pm 0.08)\) ps in PDG1996 [18], one can find the ratio

$$\begin{aligned} \tau (\Lambda _{b}^{0})/\tau (B^{0})=0.75\pm 0.05. \end{aligned}$$
(6)

Theoretically, the ratio deviating from unity at the level of \(20\%\) is considered to be too large. Nowadays, we know that the low value of \(\tau (\Lambda _{b})/\tau (B_{d})\) or the short \(\Lambda _{b}\) lifetime was a purely experimental issue. The world averages in PDG2022 are

$$\begin{aligned}{} & {} \tau (\Lambda _{b})=(1.471\pm 0.009)\ \textrm{ps},\nonumber \\ {}{} & {} \quad \tau (\Lambda _{b}^{0})/\tau (B^{0})=0.964\pm 0.007, \end{aligned}$$
(7)

which are in good agreement with the HQE prediction [5]:

$$\begin{aligned} \frac{\tau (\Lambda _{b})}{\tau (B_{d})}^{\textrm{HQE}2014}&=1-(0.8\pm 0.5)\%_{1/m_{b}^{2}}-(4.2\pm 3.3)\%_{1/m_{b}^{3}}^{\Lambda _{b}}\nonumber \\&\quad -(0.0\pm 0.5)\%_{1/m_{b}^{3}}^{B_{d}}-(1.6\pm 1.2)\%_{1/m_{b}^{4}}\nonumber \\&=0.935\pm 0.054. \end{aligned}$$
(8)

However, it can be seen from Eq. (8) that the main uncertainty in the lifetime ratio comes from the \(1/m_{b}^{3}\) corrections of the \(\Lambda _{b}\)-matrix elements. The relevant matrix elements can be parameterized in a model-independent way as follows [14]:

$$\begin{aligned} \langle \Lambda _{b}|(\bar{b}q)_{V-A}(\bar{q}b)_{V-A}|\Lambda _{b}\rangle&=f_{B_{q}}^{2}m_{B_{q}}m_{\Lambda _{b}}L_{1},\nonumber \\ \langle \Lambda _{b}|(\bar{b}q)_{S-P}(\bar{q}b)_{S+P}|\Lambda _{b}\rangle&=f_{B_{q}}^{2}m_{B_{q}}m_{\Lambda _{b}}L_{2},\nonumber \\ \langle \Lambda _{b}|(\bar{b}^{\alpha }q^{\beta })_{V-A}(\bar{q}^{\beta }b^{\alpha })_{V-A}|\Lambda _{b}\rangle&=f_{B_{q}}^{2}m_{B_{q}}m_{\Lambda _{b}}L_{3},\nonumber \\ \langle \Lambda _{b}|(\bar{b}^{\alpha }q^{\beta })_{S-P}(\bar{q}^{\beta }b^{\alpha })_{S+P}|\Lambda _{b}\rangle&=f_{B_{q}}^{2}m_{B_{q}}m_{\Lambda _{b}}L_{4}, \end{aligned}$$
(9)

where \((\bar{q}_{1}q_{2})_{V-A}\equiv \bar{q}_{1}\gamma _{\mu }(1-\gamma _{5})q_{2}\) and \((\bar{q}_{1}q_{2})_{S\pm P}\equiv \bar{q}_{1}(1\pm \gamma _{5})q_{2}\). In the literature, a \(\tilde{B}\) is usually introduced to relate \(L_{3}\) to \(L_{1}\)

$$\begin{aligned} L_{3}=-\tilde{B}\ L_{1}. \end{aligned}$$
(10)

The theoretical predictions for \(L_{1}\), \(L_{2}\) and \(\tilde{B}\) are listed in Table 1, from which one can see that there are significant differences in the theoretical predictions for \(L_{1}\) and \(L_{2}\). In this work, we intend to clarify this issue.

Table 1 \(L_{1}\), \(L_{2}\) and \(\tilde{B}\) predicted by different theoretical methods. This table is reproduced from Ref. [5]
Full size table

In Ref. [24], we performed a QCD sum rules analysis of the weak decay form factors of doubly heavy baryons to singly heavy baryons. However, considering that there were few theoretical predictions and experimental data available to compare with, we then applied our calculation method to study the semileptonic decay of \(\Lambda _{b}\rightarrow \Lambda _{c}l\bar{\nu }\) in Ref. [25]. It turned out that our predictions for the form factors and decay widths were consistent with those of heavy quark effective theory (HQET) and lattice QCD. In this work, similar computing strategies are undertaken. At the QCD level, contributions from up to dimension-6 four-quark operators are considered to obtain the matrix elements in Eq. (9).

The remainder of this article is organized as follows. In Sect. 2, the main steps of QCD sum rules are presented, and in Sect. 3, numerical results are shown. For consistency, the pole residue of \(\Lambda _{b}\) is also considered. We conclude this article in the last section.

2 QCD sum rules analysis

The following interpolating current is adopted for \(\Lambda _{b}\):

$$\begin{aligned} J=\epsilon _{abc}(u_{a}^{T}C\gamma _{5}d_{b})Q_{c}, \end{aligned}$$
(11)

where Q denotes the bottom quark, a, b, c are color indices, and C is the charge conjugation matrix. The correlation function is defined as

$$\begin{aligned} \Pi (p_{1},p_{2})= & {} i^{2}\int \textrm{d}^{4}x\textrm{d}^{4}y\ e^{-ip_{1}\cdot x+ip_{2}\cdot y}\langle 0|T\{J(y)\Gamma _{6}(0)\bar{J}(x)\}|0\rangle ,\nonumber \\ \end{aligned}$$
(12)

with \(\Gamma _{6}\) being a four-quark operator given in Eq. (9). Note that in spin space, J(y), \(\Gamma _{6}(0)\) and \(\bar{J}(x)\) are \(4\times 1\), \(1\times 1\), and \(1\times 4\) matrices, respectively, and therefore \(\Pi (p_{1},p_{2})\) is a \(4\times 4\) matrix.

Following the standard procedure of QCD sum rules, one can calculate the correlation function at the hadron and QCD levels. At the hadron level, after inserting the complete set of baryon states, one can obtain

(13)

where \(\lambda _{H}=\lambda _{\Lambda _{b}}\), \(M=m_{\Lambda _{b}}\) are respectively the pole residue and mass of \(\Lambda _{b}\), the parameters a and b are introduced to parameterize the hadronic matrix element

$$\begin{aligned} \langle \Lambda _{b}(q^{\prime },s^{\prime })|\Gamma _{6}|\Lambda _{b}(q,s)\rangle =\bar{u}(q^{\prime },s^{\prime })(a+b\gamma _{5})u(q,s), \end{aligned}$$
(14)

and the ellipsis stands for the contribution from higher excited states. For the forward scattering matrix element, one can show that

$$\begin{aligned} \langle \Lambda _{b}(q,s)|\Gamma _{6}|\Lambda _{b}(q,s)\rangle&=\bar{u}(q,s)(a+b\gamma _{5})u(q,s)\nonumber \\&=2\ a\ m_{\Lambda _{b}}, \end{aligned}$$
(15)

where \(\bar{u}(q,s)u(q,s)=2\ m_{\Lambda _{b}}\) and \(\bar{u}(q,s)\gamma _{5}u(q,s)=0\) have been used. One can see that only the parameter a in Eq. (14) is relevant to the forward scattering matrix element.

It can be seen from Eq. (13) that there are eight Dirac structures, but only (at most) two parameters need to be determined. By considering the contribution of negative-parity baryons [24,25,26], one can update Eq. (13) to

(16)

Here, \(M_{+(-)}\) and \(\lambda _{+(-)}\) respectively denote the mass and pole residue of \(\Lambda _{b}(1/2^{+(-)})\), and \(a^{+-}\) is the parameter a for the positive-parity final state \(\Lambda _{b}(1/2^{+})\) and the negative-parity initial state \(\Lambda _{b}(1/2^{-})\), and so forth. When arriving at Eq. (16), we have adopted the following conventions:

$$\begin{aligned}&\langle \Lambda _{b+}(q^{\prime },s^{\prime })|\Gamma _{6}|\Lambda _{b+}(q,s)\rangle \nonumber \\&\quad =\bar{u}_{+}(q^{\prime },s^{\prime })(a^{++}+b^{++}\gamma _{5})u_{+}(q,s),\nonumber \\&\langle \Lambda _{b+}(q^{\prime },s^{\prime })|\Gamma _{6}|\Lambda _{b-}(q,s)\rangle \nonumber \\&\quad =\bar{u}_{+}(q^{\prime },s^{\prime })(a^{+-}+b^{+-}\gamma _{5})(i\gamma _{5})u_{-}(q,s),\nonumber \\&\langle \Lambda _{b-}(q^{\prime },s^{\prime })|\Gamma _{6}|\Lambda _{b+}(q,s)\rangle \nonumber \\&\quad =\bar{u}_{-}(q^{\prime },s^{\prime })(i\gamma _{5})(a^{-+}+b^{-+}\gamma _{5})u_{+}(q,s),\nonumber \\&\langle \Lambda _{b-}(q^{\prime },s^{\prime })|\Gamma _{6}|\Lambda _{b-}(q,s)\rangle \nonumber \\&\quad =\bar{u}_{-}(q^{\prime },s^{\prime })(i\gamma _{5})(a^{--}+b^{--}\gamma _{5})(i\gamma _{5})u_{-}(q,s). \end{aligned}$$
(17)

In Eq. (17), these \(i\gamma _{5}\) are not necessary, but they are convenient.

At the QCD level, the correlation function can be written formally as

(18)

The coefficients \(A_{i}\) are then expressed as double dispersion relations

$$\begin{aligned} A_{i}(p_{1}^{2},p_{2}^{2},q^{2})\!=\!\int ^{\infty }\textrm{d}s_{1}\int ^{\infty }\textrm{d}s_{2}\frac{\rho ^{A_{i}}(s_{1},s_{2},q^{2})}{(s_{1}\!-\!p_{1}^{2})(s_{2}-p_{2}^{2})},\nonumber \\ \end{aligned}$$
(19)

where the spectral function \(\rho ^{A_{i}}(s_{1},s_{2},q^{2})\) can be calculated using Cutkosky cutting rules, see Fig. 1. Sum rules are obtained by equating Eq. (16) with Eq. (18) and then using quark–hadron duality to eliminate the contribution of excited states. Furthermore, by equating the coefficients of the same Dirac structure, one can have eight equations to solve eight unknown parameters \(a^{\pm \pm }\) and \(b^{\pm \pm }\). In particular, after performing the Borel transform, one can arrive at

$$\begin{aligned} a^{++}= & {} \frac{\{M_{-}^{2},M_{-},M_{-},1\}.\{\mathcal{B}A_{1},\mathcal{B}A_{2},\mathcal{B}A_{3},\mathcal{B}A_{4}\}}{\lambda _{+}^{2}(M_{+}+M_{-})^{2}}\nonumber \\{} & {} \exp \left( \frac{2M_{+}^{2}}{T^{2}}\right) , \end{aligned}$$
(20)

where \(\mathcal{B}A_{i}\) are doubly Borel-transformed coefficients

$$\begin{aligned} \mathcal{B}A_{i}=\int ^{s_{0}}\textrm{d}s_{1}\int ^{s_{0}}\textrm{d}s_{2}\ \rho ^{A_{i}}(s_{1},s_{2},q^{2})\exp \left( -\frac{s_{1}+s_{2}}{T^{2}}\right) ,\nonumber \\ \end{aligned}$$
(21)

and \(s_{0}\) and \(T^{2}\) are the continuum threshold parameter and the Borel parameter, respectively.

Fig. 1
figure 1

Perturbative contribution. Cutting rules are also shown

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Fig. 2
figure 2

All nonzero, “independent” diagrams considered in this work

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In this work, contributions from up to dimension-6 four-quark operators are considered at the QCD level.Footnote 1 For the matrix elements in Eq. (9), we find that contributions from quark condensate (dimension-3) and quark-gluon condensate (dimension-5) are proportional to the mass of the u/d quark, which is taken to be zero in this work. All nonzero, “independent” diagrams are shown in Fig. 2. Here, “independent” implies non-equivalence. For example, diagram dim-4-2,5 with quark 2 and quark 5 each emitting a gluon is equal to diagram dim-4-1,4; therefore, the former is not listed in Fig. 2.

2.1 The pole residue

As can be seen in Eq. (20), the pole residue of \(\Lambda _{b}\) is an indispensable input. For consistency, in this work, we also perform an analysis on the pole residue of \(\Lambda _{b}\), whose sum rule is [25]

$$\begin{aligned}{} & {} (M_{+}+M_{-})\lambda _{+}^{2}\exp (-M_{+}^{2}/T_{+}^{2})\nonumber \\{} & {} \quad =\int ^{s_{0}}\textrm{d}s\ (M_{-}\rho ^{A}+\rho ^{B})\exp (-s/T_{+}^{2}), \end{aligned}$$
(22)

from which, one can obtain the mass formula for \(\Lambda _{b}\)

$$\begin{aligned} M_{+}^{2}=\frac{\int ^{s_{0}}\textrm{d}s\ (M_{-}\rho ^{A}+\rho ^{B})\ s\ \exp (-s/T_{+}^{2})}{\int ^{s_{0}}\textrm{d}s\ (M_{-}\rho ^{A}+\rho ^{B})\exp (-s/T_{+}^{2})}. \end{aligned}$$
(23)

Equation (23) can be viewed as a constraint to Eq. (22). Following Ref. [27], in this work, we use Eq. (23) to determine the continuum threshold parameter \(s_{0}\). Specifically, for a set of fixed parameters (renormalization scale, condensate parameters, etc.), the optimal \((s_{0}, T_{+}^{2})\) are obtained through the following procedure:

  1. 1.

    For a trial \(s_{0}\), plot the pole residue curve with respect to the Borel parameter \(T_{+}^{2}\) using Eq. (22). Find the minimum point \(T_{+}^{2}\) on the curve (see Fig. 3 for some intuitive impressions).

  2. 2.

    Substitute the set of (\(s_{0}\), \(T_{+}^{2}\)) obtained in step 1 into Eq. (23) to calculate the baryon mass, and compare it with the experimental value. If equal (within a small error range), terminate; otherwise, go to step 1.

For different input parameters, one can obtain different optimal (\(s_{0}\), \(T_{+}^{2}\)), which are listed in Table 2.

Fig. 3
figure 3

The pole residue \(\lambda _{\Lambda _{b}}\) as a function of the Borel parameter \(T_{+}^{2}\). The red, orange and blue dots respectively correspond to the renormalization scales \(\mu =m_{b}\), \(\mu =6\ \textrm{GeV}\) and \(\mu =3\ \textrm{GeV}\). The minimum points are marked, and they correspond to the experimental mass of \(\Lambda _{b}\) via Eq. (23)

Full size image

In this work, contributions from up to dimension-6 four-quark operators are also considered for the sum rule in Eq. (22).

3 Numerical results

3.1 Inputs

Our main inputs in numerical calculation can be found in Table 2. We take the \(\overline{\textrm{MS}}\) mass for the bottom quark [17] and neglect the mass of the u/d quark. The condensate parameters are taken from Ref. [28]. The renormalization scale is taken as \(\mu _{b}=3{-}6\ {\textrm{GeV}}\), with \(m_{b}(m_{b})\) the central value [27], from which one can estimate the dependence of the calculation results on the renormalization scale.

3.2 The pole residue and the continuum threshold parameter

Our predictions for the pole residue \(\lambda _{\Lambda _{b}}\), together with the continuum threshold parameter \(s_{0}\), can be found in Table 2. A similar investigation was also performed in our previous work [25]; however, in this work, more contributions from higher-dimensional operators are considered. Numerically, the predictions in this work are close to those in Ref. [25]. This is essentially because the contributions from higher-dimensional operators are small. A comprehensive study of the pole residues of anti-triplet heavy baryons can be found in Ref. [29].

Table 2 Our predictions of the pole residue \(\lambda _{\Lambda _{b}}\), \(L_{1,2,3,4}\) defined in Eq. (9), and \(\tilde{B}\equiv -L_{3}/L_{1}\). For a set of fixed parameters (renormalization scale, condensate parameters, etc.), the continuum threshold parameter \(s_{0}\) can be determined, and then the quantities we are interested in can also be determined—by requiring them to have as little dependence on the Borel parameter as possible
Full size table

3.3 The four-quark operator matrix elements

The four-quark operator matrix elements in Eq. (9) are all proportional to some parameter \(L_{i}\). In Fig. 4, we plot the curves of \(L_{1,2,3,4}\) as functions of the Borel parameter \(T^{2}\), on which the stability region can be found. In these figures, we have also evaluated the dependence of \(L_{1,2,3,4}\) on the renormalization scale. The corresponding results are summarized in Table 2.

Adding all the uncertainties from various input parameters in quadrature, our final results of \(L_{1,2,3,4}\) and \(\tilde{B}\equiv -L_{3}/L_{1}\) are respectively

$$\begin{aligned} L_{1}&=-0.143\pm 0.028,\nonumber \\ L_{2}&=+0.0746\pm 0.0142,\nonumber \\ L_{3}&=+0.151\pm 0.027,\nonumber \\ L_{4}&=-0.0764\pm 0.0139, \end{aligned}$$
(24)

and

$$\begin{aligned} \tilde{B}=1.057\pm 0.030, \end{aligned}$$
(25)

where \(m_{B_{q}}=5.280\ \textrm{GeV}\) and \(f_{B_{q}}=186\ \textrm{MeV}\) [14] have been used.

Fig. 4
figure 4

\(L_{1,2,3,4}\) as functions of the Borel parameter \(T^{2}\)

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Some comments are in order.

  • The spectral density in Eq. (19) also depends on \(q^{2}\equiv (p_{1}-p_{2})^{2}\). For the forward scattering matrix elements of interest, \(q^{2}\) is taken to be 0.

  • In Eq. (9), the only difference between the first matrix element \(\langle \Lambda _{b}|(\bar{b}q)_{V-A}(\bar{q}b)_{V-A}|\Lambda _{b}\rangle \) and the third matrix element \(\langle \Lambda _{b}|(\bar{b}^{\alpha }q^{\beta })_{V-A}(\bar{q}^{\beta }b^{\alpha })_{V-A}|\Lambda _{b}\rangle \) is in the color space. For all the diagrams in Fig. 2 except for diagrams (dim-4-1,4), (dim-4-1,5) and (dim-4-2,4), the color factors for the two matrix elements differ only by a negative sign. Considering that the contributions of these three gluon condensate diagrams are small, \(\tilde{B}\equiv -L_{3}/L_{1}\) only deviates slightly by 1.

  • One can see from Table 2 that the uncertainty caused by \(\mu \) dependence is dominant, which is about \(20\%\). As a comparison, in Ref. [27], the authors performed a QCD sum rules analysis on the decay constants of B and \(B_{s}\) mesons. \(\mu \) is also taken in the range of 3–6 GeV with \(m_{b}(m_{b})\) as the central value. For the perturbative spectral function, contributions from up to \(\alpha _{s}^{2}\) order are considered. Scale dependence is also the main source of error, which is approximately \(10\%\). It can be expected that when higher-order corrections are considered, the scale dependence should decrease.

  • In Fig. 4, we have also shown the calculation errors, which are essentially small. Nevertheless, when we try to find the stability region on the curve, these small errors also play a role.

  • As can be seen in Table 2, for a set of fixed parameters (quark mass, renormalization scale, condensate parameters), the continuum threshold parameter \(s_{0}\) can be determined, and then the quantities we are interested in can also be determined—by requiring them to have as little dependence on the Borel parameter as possible.

  • In our opinion, the continuum threshold parameter \(s_{0}\) is the most important parameter in QCD sum rules. In fact, once \(s_{0}\) is fixed, the quantity that we are interested in is almost determined by searching for the stability region. In the literature, there exist at least two approaches for determining \(s_{0}\). One approach is to empirically select \(\sqrt{s_{0}}\) approximately 0.5 GeV larger than the ground state mass. Another approach is to determine \(s_{0}\) through the mass formula. We have adopted the latter approach in this work. The basic logic behind doing so is that the mass formula can be seen as a constraint to the sum rule of the two-point correlation function. Of course, this comes with the cost of abandoning the predictive power of hadron mass.

4 Conclusions

Heavy quark expansion can nicely explain the lifetime of \(\Lambda _{b}\). However, there still exist sizable uncertainties in the four-quark operator matrix elements of \(\Lambda _{b}\) in \(1/m_{b}^{3}\) corrections, which describe the spectator effects. In this work, these four-quark operator matrix elements are investigated using full QCD sum rules. At the QCD level, contributions from up to dimension-6 four-quark operators are considered. A stable Borel region can be found. We have also considered the uncertainties from various input parameters, and find that the main source of error is scale dependence, which is about 20%. Our method for calculating high-dimensional operator matrix elements holds promise for use in resolving the \(\Omega _{c}\) lifetime puzzle.